M. Renardy [1]} established the local well-posedness in Sobolev spaces with potential \(\mathcal {U}(R)=(1-|R|^2)^{1-\sigma }\)  for \(\sigma >1\) . Later, B. Jourdain, T. Lelièvre, and
C. Le Bris [2]} proved local existence of a stochastic differential equation with potential \(\mathcal {U}(R)=-k\log (1-|R|^{2})\)  in the case \(k>3\)  for a Couette flow. H. Zhang and P. Zhang [3]} proved local well-posedness of (1.4) with \(d=3\)  in weighted Sobolev spaces. For the co-rotation case, F. Lin, P. Zhang, and Z. Zhang [4]} obtained a global existence results with \(d=2\)  and \(k > 6\) . If the initial data is perturbation around equilibrium, N. Masmoudi [5]} proved global well-posedness of (1.4) for \(k>0\) . In the co-rotation case with \(d=2\) , he [5]} obtained a global result for \(k>0\)  without any small conditions. In the co-rotation case, A. V. Busuioc, I. S. Ciuperca, D. Iftimie and L. I. Palade [7]} obtained a global existence result with only the small condition on \(\psi _0\) . The global existence of weak solutions in \(L^2\)  was proved recently by N. Masmoudi [8]} under some entropy conditions.
Recently, M. Schonbek [9]} studied the \(L^2\)  decay of the velocity for the co-rotation
FENE dumbbell model, and obtained the
decay rate \((1+t)^{-\frac{d}{4}+\frac{1}{2}}\) , \(d\ge 2\)  with \(u_0\in L^1\) .
Moreover, she conjectured that the sharp decay rate should be \((1+t)^{-\frac{d}{4}}\) , \(d\ge 2\) .
However, she failed to get it because she could not use the bootstrap argument as in [10]} due to the
additional stress tensor. Recently, W. Luo and Z. Yin [11]} improved Schonbek's result
and showed that the decay rate is \((1+t)^{-\frac{d}{4}}\)  with \(d\ge 3\)  and \(\ln ^{-l}(1+t)\)  with \(d=2\)  for any \(l\in \mathbb {N^+}\) .
This result shows that M. Schonbek's conjecture is true when \(d \ge 3\) . More recently, W. Luo and Z. Yin [12]} improved the decay rate to \((1+t)^{-\frac{d}{4}}\)  with \(d=2\) .
